# Second Derivatives and Beyond Examples

#### Third Derivatives and Beyond

We're learning calculus, so, by definition, we're in the prime of our lives. Things are great. Things are awesome. We can take third derivatives, fourth derivatives, and so on. Writing f " for...

#### Critical Points

We say x = c is a critical point of the function f if f (c) exists and f'(c) = 0 or is undefined. It's generally a peak or valley in the curve. It's where the slopes becomes interesting. When...

#### Points of Inflection

A point of inflection or inflection point, abbreviated IP, is an x-value at which the concavity of the function changes. In other words, an IP is an x-value where the sign of the second derivative...

#### First Derivative Test

Sample ProblemBelow is a graph of a function f with a minimum at x = x0. Determine the sign of the derivative f' at each labelled x-value.Below is a graph of a function f with a maximum at x = x0....

#### Second Derivative Test

Sample ProblemsAssume f is defined and twice differentiable on the whole real line. Around a minimum of the function f, is f concave up or concave down? Assume f is defined and twice dif...

#### Local vs. Global Points

Sometimes we take vacations, sometimes we take stay-cations. Second derivatives can be used to determine if the function will be traveling somewhere extreme or if it will travel somewhere more subd...

#### Using Derivatives to Draw Graphs

The goal of this section is to be able to go from a formula of a function to an accurate graph of that function. We use the first and second derivatives to help find exact points on the graph and t...

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